Optimum thresholding using mean and conditional mean squared error
Abstract
Joint work with Josè E. Figueroa-Lòpez, Washington University in St. Louis
Abstract: We consider a univariate semimartingale model for (the logarithm
of) an asset price, containing jumps having possibly infinite activity. The
nonparametric threshold estimator\hat{IV}_n of the integrated variance
IV:=\int_0^T\sigma^2_sds proposed in Mancini (2009) is constructed using
observations on a discrete time grid, and precisely it sums up the squared
increments of the process when they are below a threshold, a deterministic
function of the observation step and possibly of the coefficients of X. All the
threshold functions satisfying given conditions allow asymptotically consistent
estimates of IV, however the finite sample properties of \hat{IV}_n can depend
on the specific choice of the threshold.
We aim here at optimally selecting the threshold by minimizing either the
estimation mean squared error (MSE) or the conditional mean squared error
(cMSE). The last criterion allows to reach a threshold which is optimal not in
mean but for the specific volatility and jumps paths at hand.
A parsimonious characterization of the optimum is established, which turns
out to be asymptotically proportional to the Lévy's modulus of continuity of
the underlying Brownian motion. Moreover, minimizing the cMSE enables us
to propose a novel implementation scheme for approximating the optimal
threshold. Monte Carlo simulations illustrate the superior performance of the
proposed method.